mrow> C 0 [ 0, T ] F ( ρ x ) d m ( x ) = exp { + n 2 a 2 ρ 2 } . (3.8)

Example. For the scale factor ρ = { , 1 100 , 1 10 , 1 , 10 , 10 2 , } , we can investigate the very interesting behavior of the Wiener integral:

( a ) C 0 [ 0, T ] F ( 1 100 x ) d m ( x ) = [ C 0 [ 0, T ] F ( x ) d m ( x ) ] 1 10000 ( b ) C 0 [ 0, T ] F ( 1 10 x ) d m ( x ) = [ C 0 [ 0, T ] F ( x ) d m ( x ) ] 1 100 ( c ) [ C 0 [ 0, T ] F ( 1 100 x ) d m ( x ) ] 100 = [ C 0 [ 0, T ] F ( 1 10 x ) d m ( x ) ] ( d ) C 0 [ 0, T ] F ( ρ x ) d m ( x ) = [ C 0 [ 0, T ] F ( x ) d m ( x ) ] ρ 2 (3.9)

Remark.

1) Whenever the scale factor ρ > 1 is increasing, the Wiener integral increases very rapidly. Whenever the scale factor 0 < ρ < 1 is decreasing, the Wiener integral decreases very rapidly.

2) The function G ( ρ ) = | C 0 [ 0, T ] F ( ρ x ) d m ( x ) | for F ( x ) in (3.1) is an increasing function of a scale factor ρ > 0 , because the exponential function y = e x 2 is an increasing function of x R .

3) Whenever the scale factor ρ > 0 is increasing and decreasing, the Wiener integral varies very rapidly.

4. Conclusions

What we have done in this research is that we investigate the very interesting behavior of the scale factor for the Wiener integral of an unbounded function.

From these results, we find an interesting property for the Wiener integral as a function of a scale factor which was first defined in [13] .

Note that the function in [13] is bounded and the function of this paper is unbounded!

Finally, we introduce the motivation and the application of the Wiener integral and the Feynman integral and the relationship between the scale factor and the heat (or diffusion) equation:

Remark.

1) The solution of the heat (or diffusion) equation

ψ t = i h [ h 2 2 m ψ 2 ξ 2 + V ( ξ ) ψ ] , (3.10)

is that for a real λ > 0 ,

ψ λ ( t , ξ ) = C 0 t exp { i h 0 t V ( λ 1 2 x ( s ) + ξ ) d s } ψ ( λ 1 2 x ( s ) + ξ ) d m ( s ) (3.11)

where ψ λ ( , ξ ) = ϕ ( ξ ) and ϕ L 2 ( R d ) and ξ R d and x ( ) is a R d -valued continuous function defined on [ 0, t ] such that x ( 0 ) = 0 .

2) H = Δ + V is the energy operator (or, Hamiltonian) and Δ is a Laplacian and V : R d R is a potential. This Formula (3.11) is called the Feynman-Kac formula. The application of the Feynman-Kac Formula (in various settings) has been given in the area: diffusion equations, the spectral theory of the schrödinger operator, quantum mechanics, statistical physics, for more details, see the paper [8] and the book [12] .

3) If we let λ = ρ 2 , the solution of this heat (or diffusion) equation is

ψ ρ ( t , ξ ) = C 0 t exp { i h 0 t V ( ρ x ( s ) + ξ ) d s } ϕ ( ρ x ( s ) + ξ ) d m ( s ) (3.12)

4) If we let h = m i λ = i m ρ 2 , then

ψ ρ ( t , ξ ) = C 0 t exp { + m ρ 2 0 t V ( ρ x ( s ) + ξ ) d s } ϕ ( ρ x ( s ) + ξ ) d m ( s ) (3.13)

is a solution of a heat (or diffusion) equation:

ψ t = 1 m ρ 2 [ ( m 2 ρ 2 2 m ) ψ 2 ξ 2 + V ( ξ ) ψ ] . (3.14)

This equation is of the form:

ψ t = 1 2 ψ 2 ξ 2 + 1 m ρ 2 V ( ξ ) ψ . (3.15)

5) If we let F ( x ) = exp { i h 0 t V ( λ 1 2 x ( s ) + ξ ) d s ϕ ( λ 1 2 x ( s ) + ξ ) } , then we can express the solution of the heat (or diffusion) equation by the formula

ψ ρ ( t , ξ ) = C 0 t F ( ρ x ) d m ( x ) , ψ λ ( t , ξ ) = C 0 t F ( λ 1 2 x ) d m ( x ) (3.16)

6) By this motivation, we first define the scale factor of the Wiener integral by the real number ρ > 0 in the paper [13] .

Remark. I am very grateful for the referee to comment in details.

Supported

This article was supported by the National Research Foundation grant NRF-2017R1A6311030667.

Cite this paper
Kim, Y. (2019) Behavior of a Scale Factor for Wiener Integrals of an Unbounded Function. Applied Mathematics, 10, 326-332. doi: 10.4236/am.2019.105023.
References

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https://doi.org/10.4236/am.2018.95035

 
 
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